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Interior points

  1. Jul 10, 2009 #1
    Hallo,

    My teacher wrote that:

    "The set has no interior points, and neither does its complement, R\Q" where R refers real

    numbers and Q is the rationals numbers.

    why can't i find an iterior point?

    thanks,

    Omri
     
  2. jcsd
  3. Jul 10, 2009 #2
    Let q be an arbitrary rational number. Does there exist a neighborhood of q that is a subset of Q?
     
  4. Jul 10, 2009 #3
    How about the fact that I can squeeze a real number between any two arbitrary points of Q?
     
  5. Jul 10, 2009 #4

    HallsofIvy

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    Irrelevant. Do you mean an irrational number? Now that would be relevant.
     
  6. Jul 10, 2009 #5

    HallsofIvy

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    So "the set" is Q. For p to be an interior point of Q, there must exist an interval around p, [math](p-\delta, p+\delta)[/quote] consisting entirely of rational numbers. For p to be an interior point of R\Q, the set of irrational numbers, there must exist an interval [itex](p- \delta, p+ \delta)[/itex]] consisting entirely of irrational numbers. There is NO interval of real numbers consisting entirely of rational number or entirely of irrational numbers.
     
    Last edited: Jul 11, 2009
  7. Jul 10, 2009 #6
    That was what I said anyway, but of course a real is not necessarily rational part got lost along the way... Sorry for that.
     
  8. Jul 11, 2009 #7
    thank you for your comments,

    I'm sorry but the statement (as i guess you already assume) was:

    "The set Q has no interior points, and neither does its complement, R\Q"

    thanks

    Omri
     
  9. Jul 11, 2009 #8

    HallsofIvy

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    Yes, that was essentially what everyone was assuming.
     
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